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Comparison Among Groups with Francis Parameterization

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In my last post, I suggested that the Francis parameterization of the von Bertalanffy growth model may be used in cases where the typical parameterization did not converge (likely due to issues related to highly correlated parameters and data with lengths that are highly variable within ages and the full curvature of the model is not readily apparent because the data are truncated for some reason (e.g., high mortality rates, size-selective gear)). A follow-up question to that post is how to compare parameter estimates between sexes when using the Francis parameterization. This is largely the same as the demonstration for the typical parameterization in the Von Bertalanffy Growth – Intro Vignette on the fishR page except, of course, that the user must write out the more bulky Francis parameterization. This post is a demonstration of the required code (with few comments as most of this is generally described in the Von Bertalanffy Growth – Intro Vignette).

Thank You

I would like to fish publicly thank Yihui Xie, the author of the knitr package for his quick response to my request for an additional feature to his knit2wp() function which allows one to easily create blog posts such as this on wordpress.com in RStudio using the markdown language. The knitr package is amazing, but Yihui’s attention to his users is even more amazing. Thanks Yihui!

Preliminaries and Data Manipulation

Begin by loading the FSA package, …

library(FSA)

the data (note that the working direction would have been set before read.csv()), …

From this it is seen that there is no difference in the or parameters but there may be in the parameter. The model with in common (i.e., fit12) fits slightly better (lower RSS) then the model with in common, so the following tests will compare the two two parameter in common models that also have in common to the model with only in common …

These results suggest that differs between sexes and, perhaps, that also differs between sexes depending on the level of that one is using. It is probably reasonable to use an of 0.1 with this type of data because of the high degree of variability in lengths and the likely interest in whether there is even a slight difference between sexes. Of course, should have been set way before this stage (i.e., way before looking at the results). I will continue assuming that was 0.05 and that these results show that only differs between sexes.

Now compare the model with and , but not , in common to the model with no differences between sexes to confirm that differs between the sexes …

This model suggests that all parameters in common is at least as good of a model as the model where differs. This, however, is inconsistent with what was seen above. Perhaps the inconsistency comes from the fact that both and should be allowed to differ. Thus, try comparing the model with separate and parameters to the model with all parameters in common.

Thus, the AIC results suggest (though not strongly) that the mean length of males and females differs at age-8.5, potentially differ at age-5, but do not differ at age-12. The coefficient results from the best-fit model(s) give an indication of the difference in mean lengths at

Notes

I am not sure what to do with the two “older” females in the sample. It does not seem correct to the fit both models out to age-19 as there are no males older than age-11 and only the two females older than age-12. I decided to simply fit the model over the general ages where both sexes were observed.

I use the two final plots above to demonstrate the results of the models. If I was preparing these plots for publication I would not constrain the two lines to have the exact same parameter. In other words, I would plot the two curves from the fitGen model and then note that the parameter was not statistically different.